At the end of the last episode of We Are Lady Parts the music over the credits, I eventually discovered, was Yaar Vekho by Sanam Marvi. I have no idea what the devotional is about but I listen to it pretty regularly.
Prisencolinensinainciusol is an absolute jam, but I don’t know if it counts given that it’s in a language nobody understands. Supposedly it’s supposed to be what American English sounds like to Europeans.
For languages that I don’t know at all, Ue o Muite Arukō, definitely. As a part-time opera singer, I like a lot of songs in German, Italian, French, and Latin, all languages that I know a little, but not very well, but Japanese is almost entirely alien to me.
I do know that for Japanese people, at least of the era, although the words are literally about something else, there is a subtext something like, “OK, we lost the war. But we have to pick ourselves up, because Life goes on.”
Most songs on Najma’s album Qareeb
"Chordhiya"on Susheela Rahman’s Music for Crocodiles
https://www.google.com/search?q=Chordhiya&client=ms-android-verizon&sourceid=chrome-mobile&ie=UTF-8
Most songs on Dengue Fever’s Debgue Fever
It’s probably impossible for me to choose and absolute favorite—too many possibilities, jostling for position, none of them I’ve ever thought to rank, formally.
But I can at least name the two that came to mind first.
#1: Billy ze Kick’s 1997 cover of “Non, Non Rien n’a changé.”
(This music video is, also, the most 1997 thing I’d seen since 1997.)
Second, if a bit cheaty…John Barry’s main titles to The Last Valley.
…taking it’s lyrics from Andreas Gryphius 1636 poem Tränen des Vaterlandes, about the horrors of the Thirty Years War.
Not exactly something you can dance to, admittedly.
Does gibberish count as a language? If so, then my vote goes to “Prisencolinensinainciusol” by Adriano Celentano.
It was an experiment to see if a song could be a hit in Italy with gibberish lyrics that sound English. And, listening to it, you can pick out little bits where you can almost understand what the word is. But you can’t.
There’s very little I can understand of The Cocteau Twins but I’ve always loved them all the same, the lyrics are idiosyncratic english mostly though indecipherable so that perhaps doesn’t count.
Same for Kevin Rowland, what the hell is he talking about here? no-one will ever know.
I do like “pulse” by the mad capsule markets, again, it is mangled english but I think it is foreign enough to count
for strictly foriegn language I very much like “links” by Rammstein, great video too.
and at the other end of the scale, “mais si tu dois partir” by Fairport Convention
There are so many.
Te Vaka a group from New Zealand
Heilung, Danheim from Scandinavia
Otyken Siberian natives
I know a little French (I took it for a year in high school), but buggered if I can understand a word of what she’s singing:
I often like the opening/closing music in anime – but I can’t think of a specific example.
Brian
Back in the '90s, I was co-hosting a drive-in radio show with a woman (American) who spoke fluent Japanese after having lived and worked in Japan.
“Sukiyaki” came up in rotation, and we thought it was so pretty we played it twice in a row with her providing a simultaneous translation the second time.
Those were fun days! 
Since its mid-90s release, I’ve loved every song on this Putumayo assemblage, so I can sing every syllable of every one — but I only know what I’m singing for the handful in Spanish. (The languages include Shona and Luganda).
While watching The Queens Gambit I discovered Tut Tut Tut Tut. Apparently that’s the sound of a busy signal.
Swedish, anyone? Not a type of music I normally listen to, but I do like this group.
This was going to me my contribution, as well. 
I completely missed you listing this. My bad. I should have +1’d it.
So many it’s hard to pick. Here are a few:
Spanish: Mecano, “Hoy No Me Puedo Levantar”
Japanese: Perfume, “Flash”
Korean: Twice, “Likey”
